I figured an equation that works for this series but have no idea how to calculate it once I have the sum...none of the rules in our book seem to apply.
I find it easier to look at these like this:
Looking at that a little bit (you can do the first few a_n if it helps to see, then you will be familiar enough to spot it right off) it should be clear that
And since you want to sum all of them, you get
At which point you have the same geometric series that TheEmptySet and Soroban solved for.